# A problem of proportion

Posted by: Gary Ernest Davis on: August 3, 2010

She was surprised to find the book apparently free of errors:

“I didn’t find any flaws in it — not in the first 15 minutes, and not even in the first hour. In fact, having used the book many times I have never found any mistakes. Not even a typo. This was disturbing. Is Richard Rusczyk human? It was such an unusual experience with an American math book, that I decided to deliberately look for a typo or a mistake.”

She wrote in considerable praise of the author’s style:

“I like this book for its amazing accuracy and clean explanations. There are also a lot of diverse problems in terms of difficulty and ideas. Richard Rusczyk has good taste. Many of the problems are from different competitions and require inventiveness. I like that there are a lot of challenge problems that go beyond the boring parts of algebra. Also, I like that important points of algebra are chosen wisely and are emphasized.”

Bu then she had an issue with the solution to problem 7.3.1 in the book. The problem is:

Five chickens can lay 10 eggs in 20 days. How long does it take 18 chickens to lay 100 eggs?

The solution given in the book, in a form shortened by Tanya, is:

The number of eggs is jointly proportional to the number of chickens and the amount of time. One chicken lays one egg in 10 days. Hence, 18 chickens will lay 100 eggs in 1000/18 days, which is slightly more than 55 and a half days.

She then remarks: “What is wrong with this solution? Richard Rusczyk is human after all.”

What information are we given?

5 chickens can lay 10 eggs in 20 days.

Assuming that chickens lay eggs at a uniform rate in appropriate units of time, and all the chickens lay at the same rate, we can use this information to figure out how many eggs 1 chicken can lay in 20 days :

1 chicken can lay 2 eggs in 20 days.

or, to put it another way:

1 chicken lays 1 egg in 10 days.

So,

18 chickens lay 18 eggs in 10 days.

To get 100 eggs from 18 chickens we have to know how many lots of 18 there are in 100. The answer is, of course, not a whole number: $\frac{100}{18}=\frac{50}{9}=5\frac{5}{9}$.

If we could had knowledge about how many eggs chickens laid in fractions of a day then we could say that:

18 chickens lay 100 eggs in $10 \times 5 \frac{5}{9} = 55 \frac{5}{9}$ days.

But we do not have such information: we do not even know how many eggs 1 chicken lays in 1 day  – $\frac{1}{10}^{\textrm{th}}$ of an egg? – the point being that while the days are divisible into fractional parts, the eggs are not, at least not as the chicken lays them (only afterward, when we make an omelet) . So we can only operate in units of 10 days, because that’s how long it takes a chicken to lay an egg. So, we can only say that:

18 chickens lay 90 eggs in 50 days, and 108 eggs in 60 days.

So, to get 100 eggs from 18 chickens we are going to have to wait 60 days –  at which point we will have 8 more eggs than the 100 we wanted.

The moral: fractional answers make sense only when the units of measurement can be divided arbitrarily. An egg, as it is being laid, is 1 egg. A chicken does not lay $\frac{1}{2}$ an egg or $\frac{1}{10}$ an egg.

### 2 Responses to "A problem of proportion"

There is nothing in the problem statement to suggest that all chickens lay on the same schedule. Your analysis depends on the implied assumption that every chicken lays an egg on day n where n ≡ 0 (mod 10). A more reasonable assumption might be that the chickens’ laying schedules are uniformly randomly distributed. An assumption of a uniform (not random) distribution leads to the answer given in the text.

Of course, there’s much more that we don’t know Do chickens lay at night? Is the interval between eggs laid by the same chicken exactly 10 days, or does it fluctuate? What are we going to do once we have 100 eggs, and is it something we can do at an arbitrary time (like 5/9 of the way through a day)? These fall, along with your objection, into the general category of “know what’s going on” — rather than regard a story problem as simply a framing of an abstract math problem, we should try to actually understand the situation that has been described.

I agree there’s nothing in the problem statement to suggest that all chickens lay on the same schedule, except that they lay an egg somewhere in 10 days.

I think that I am not implicitly assuming every chicken lays an egg on day n where n ≡ 0 (mod 10): I simply do not know where in a 10 day period a chicken lays an egg. We are not told that – only that at some point in a 10 day period a chicken will lay an egg.

It might indeed be more reasonable to assume the chickens’ laying schedules are uniformly randomly distributed, but that’s an assumption added to the problem statement, and is not part of how the problem was posed.